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Numbers k such that A246601(k) > 2*k.
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4095, 8190, 16380, 32760, 65520, 131040, 262080, 524160, 1048320, 2096640, 4193280, 8386560, 16773120, 16777215, 33546240, 33550335, 33554430, 67092480, 67096575, 67100670, 67108860, 134184960, 134189055, 134193150, 134201340, 134217720, 268369920, 268374015
COMMENTS
An analog of abundant numbers k ( A005101), in which the divisor sum is restricted to divisors d whose 1-bits in their binary expansions are common with those of k. If k is a term then 2*k is also a term. Therefore all the terms can be generated from the primitive set of the odd terms ( A359085). The least term that is not divisible by 4095 is a(208) = 1099511627775 = 2^40 - 1.
Since A246601(2^k-1) = sigma(2^k-1), 2^k-1 is a term for all k in A103292, unless 2^k-1 is an odd perfect number ( A000396).
MATHEMATICA
s[n_] := DivisorSum[n, # &, BitAnd[n, #] == # &]; Select[Range[10^6], s[#] > 2*# &]
PROG
(PARI) is(n) = sumdiv(n, d, d * (bitor(n, d) == n)) > 2*n;
Odd numbers k such that A246601(k) > 2*k.
+20
2
4095, 16777215, 33550335, 67096575, 134189055, 268374015, 536743935, 1073483775, 2146963455, 4293922815, 8587841535, 17175678975, 34351353855, 68702703615, 68719476735, 137405403135, 137422176255, 137438949375, 274810802175, 274827575295, 274844348415, 274877894655
COMMENTS
These are the odd terms of A359084 and also its primitive terms, since if m is a term then m*2^k is a term of A359084 for all k >= 0. The least term that is not divisible by 4095 is a(29) = 1099511627775 = 2^40 - 1.
MATHEMATICA
s[n_] := DivisorSum[n, # &, BitAnd[n, #] == # &]; Select[Range[1, 2^24, 2], s[#] > 2*# &]
PROG
(PARI) is(n) = n%2 && sumdiv(n, d, d * (bitor(n, d) == n)) > 2*n;
Number of divisors d of n with property that the binary representation of d can be obtained from the binary representation of n by changing any number of 1's to 0's.
+10
13
1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 2, 2, 4, 3, 2, 2, 2, 2, 4, 2, 2, 6, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2, 4, 2, 3, 2, 2, 4, 2
COMMENTS
Equivalently, the number of divisors d of n such that the bitwise OR of n and d is equal to n. - Chai Wah Wu, Sep 06 2014 Equivalently, the number of divisors d of n such that the bitwise AND of n and d is equal to d. - Amiram Eldar, Dec 15 2022 The sums of the first 10^k terms for k = 1, 2, ..., are 16, 224, 2580, 26920, 273407, 2745100, 27440305, 274127749, 2738936912, 27373288534, 273631055291, 2735755647065, ... . Conjecture: The asymptotic mean of this sequence is 1 + Sum_{k>=1} 1/(k*2^ A000120(k)) = 2.7351180693... . - Amiram Eldar, Apr 07 2023
FORMULA
a(2^i) = 1.
a(odd prime) = 2.
a(n) <= 2^wt(n)-1, where wt(n) = A000120(n). a(2*n) = a(n), and therefore a(m*2^k) = a(m) for m odd and k>=0.
EXAMPLE
12 = 1100_2; only the divisors 4 = 0100_2 and 12 = 1100_2 satisfy the condition, so(12)=2.
15 = 1111_2; all divisors 1,3,5,15 satisfy the condition, so a(15)=4.
MAPLE
local a, d, s, t, i, sw;
a:=0;
s:=convert(n, base, 2);
for d in numtheory[divisors](n) do
sw:= false;
t:=convert(d, base, 2);
for i from 1 to nops(t) do
if t[i]>s[i] then
sw:= true;
fi;
od:
if not sw then
a:=a+1;
fi;
od;
a;
end;
MATHEMATICA
a[n_] := DivisorSum[n, Boole[BitOr[#, n] == n]&]; Array[a, 100] (* Jean-François Alcover, Dec 02 2015, adapted from PARI *)
PROG
(Python)
from sympy import divisors
return sum(1 for d in divisors(n) if n|d == n)
Numbers k such that the set of divisors {d | k, BitOr(k, d) = k} has an integer harmonic mean.
+10
3
1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 45, 48, 56, 60, 64, 90, 96, 112, 120, 128, 180, 192, 224, 240, 256, 360, 384, 448, 480, 496, 512, 720, 768, 896, 960, 992, 1024, 1440, 1536, 1792, 1920, 1984, 2048, 2880, 3072, 3584, 3840, 3968, 4096, 5760, 6144, 7168, 7680
COMMENTS
Equivalently, the set of divisors can be defined by {d | k, BitAnd(k, d) = d}.
Analogous to harmonic (or Ore) numbers ( A001599) where the divisors d of k are restricted by BitOr(k, d) = k or BitAnd(k, d) = d. If k is a term then so is 2*k. The primitive terms are in A362805. Thus, this sequence includes all the powers of 2 ( A000079), all the numbers of the form 3*2^m and 15*2^m for m >= 1, and all the numbers of the form 7*2^m for m >= 2. All the even perfect numbers ( A000396) are terms: if k = 2^(p-1)*(2^p-1) is a perfect number (where p is a Mersenne exponent, A000043), then the only divisors of k such that BitOr(k, d) = k are 2^(p-1) and k itself, and the harmonic mean of 2^(p-1) and 2^(p-1)*(2^p-1) is 2^p - 1. Are 1 and 45 the only odd terms in this sequence?
MATHEMATICA
q[n_] := IntegerQ[HarmonicMean[Select[Divisors[n], BitAnd[n, #] == # &]]]; Select[Range[10^4], q]
PROG
(PARI) div(n) = select(x->(bitor(x, n) == n), divisors(n));
is(n) = {my(d = div(n)); denominator(#d/sum(i = 1, #d, 1/d[i])) == 1; }
a(n) is the sum of the divisors d of 2*n such that the binary expansions of d and 2*n have no common 1-bit.
+10
2
1, 3, 1, 7, 6, 6, 1, 15, 10, 13, 1, 16, 1, 3, 1, 31, 18, 33, 1, 32, 22, 3, 1, 36, 6, 3, 10, 14, 1, 6, 1, 63, 34, 54, 1, 70, 38, 22, 1, 70, 42, 48, 1, 7, 6, 3, 1, 76, 1, 38, 18, 7, 1, 24, 1, 36, 1, 3, 1, 21, 1, 3, 1, 127, 84, 116, 1, 126, 70, 38, 1, 153, 74, 77
COMMENTS
Odd numbers share a 1-bit (2^0) with all their divisors, hence this sequence deals with even numbers.
FORMULA
a(n) <= A346878(n) with equality iff n is a power of 2.
EXAMPLE
For n = 6:
- the divisors of 12 are:
d bin(d) common bit?
-- ------ -----------
1 1 no
2 10 no
3 11 no
4 100 yes
6 110 yes
12 1100 yes
- hence a(6) = 1 + 2 + 3 = 6.
MATHEMATICA
a[n_] := DivisorSum[2n, #*Boole[BitAnd[#, 2n] == 0] &]; Array[a, 74]
PROG
(PARI) a(n) = sumdiv(2*n, d, if (bitand(2*n, d)==0, d, 0))
(Python)
from sympy import divisors as divs
def a(n): return sum(d for d in divs(2*n, generator=True) if (d>>1)&n == 0)
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